Question

A student solved the following equation using the following steps:

4(2−3x)=x−2(2x+1)
8−3x=x−4x−2
8−3x=−3x−2

Answer 1

Based on the student's work, the equation was solved

Answer 2

The equation solved correctly would show that it has
solution(s).

Answer 3

Let's check the student's steps in solving the equation \(4(2-3x) = x - 2(2x + 1)\) and verify if they are correct.

1. Start with the given equation:

\[
4(2 - 3x) = x - 2(2x + 1)
\]

2. Distribute on both sides:

- Left side:

\[
4 \cdot 2 - 4 \cdot 3x = 8 - 12x
\]

- Right side:

\[
x - 2(2x + 1) = x - (4x + 2) = x - 4x - 2 = -3x - 2
\]

Now we rewrite the equation:

\[
8 - 12x = -3x - 2
\]

3. The student's noted equation after the second step was \(8 - 3x = -3x - 2\), which is incorrect since it simplifies to:

\[
8 - 12x = -3x - 2
\]

4. Let us solve \(8 - 12x = -3x - 2\):

**Rearranging the equation:**
- Add \(12x\) to both sides:

\[
8 = 12x - 3x - 2
\]

- This simplifies to:

\[
8 = 9x - 2
\]

- Now, add \(2\) to both sides:

\[
8 + 2 = 9x
\]

\[
10 = 9x
\]

- Finally, divide both sides by \(9\):

\[
x = \frac{10}{9}
\]

So, the solution to the equation is:

\[
\boxed{\frac{10}{9}}
\]

In conclusion, the student's steps contain an error in the transformation of the equation, and the correct solution is \(x = \frac{10}{9}\).

Answer 4

Let’s clarify the student's work step by step and make sure we completely understand how the equation was supposedly solved.

The original equation given is:

\[
4(2 - 3x) = x - 2(2x + 1)
\]

The subsequent steps taken by the student were:

1. **First step: Distribution**
- The left side: \( 4(2 - 3x) = 8 - 12x \)
- The right side: \( x - 2(2x + 1) = x - (4x + 2) = x - 4x - 2 = -3x - 2 \)

This leads us to the equation:

\[
8 - 12x = -3x - 2
\]

2. **Next step: Rearranging**
Here, the student seems to have rewritten it incorrectly as \(8 - 3x = -3x - 2\). Let's continue solving from where they should have correctly left off:

So, from \( 8 - 12x = -3x - 2 \), we add \(12x\) and \(2\) to isolate terms:

\[
8 + 2 = 12x - 3x
\]

Which simplifies to:

\[
10 = 9x
\]

Dividing both sides by \(9\) gives:

\[
x = \frac{10}{9}
\]

### Conclusion:

**Final answer:** The correct solution to the equation \(4(2 - 3x) = x - 2(2x + 1)\) is:

\[
\boxed{\frac{10}{9}}
\]

While the student's approach had a misstep in rewriting the equation, the resultant final solution can still be derived correctly from the initial equations as shown. If you would like to discuss or analyze further parts of the solution, please let me know!

Answer 5

Let's examine the original equation step by step to check if it has solutions and to clarify any confusion on the procedure.

### Original Equation:
\[
4(2 - 3x) = x - 2(2x + 1)
\]

### Step 1: Distribute on both sides
**Left Side:**
\[
4(2 - 3x) = 8 - 12x
\]

**Right Side:**
\[
x - 2(2x + 1) = x - (4x + 2) = x - 4x - 2 = -3x - 2
\]

### Step 2: Setting the equation
Now we have:
\[
8 - 12x = -3x - 2
\]

### Step 3: Rearranging the equation
To compare terms, rearranging gives:
\[
8 + 2 = -3x + 12x
\]
This simplifies to:
\[
10 = 9x
\]

### Step 4: Solve for \(x\)
Dividing both sides by \(9\):
\[
x = \frac{10}{9}
\]

### Conclusion on Solutions
This shows that the equation does indeed have a solution:

\[
\boxed{\frac{10}{9}}
\]

### Verification
To verify that this solution works, we can substitute \(x = \frac{10}{9}\) back into the original equation:

**Left Side:**
\[
4\left(2 - 3\left(\frac{10}{9}\right)\right) = 4\left(2 - \frac{30}{9}\right) = 4\left(\frac{18 - 30}{9}\right) = 4\left(\frac{-12}{9}\right) = 4\left(-\frac{4}{3}\right) = -\frac{16}{3}
\]

**Right Side:**
\[
\frac{10}{9} - 2\left(2\left(\frac{10}{9}\right) + 1\right) = \frac{10}{9} - 2\left(\frac{20}{9} + 1\right) = \frac{10}{9} - 2\left(\frac{20 + 9}{9}\right) = \frac{10}{9} - 2\left(\frac{29}{9}\right) = \frac{10}{9} - \frac{58}{9} = -\frac{48}{9} = -\frac{16}{3}
\]

Both sides match, confirming that \(x = \frac{10}{9}\) is indeed the solution.

### Summary
The equation \(4(2 - 3x) = x - 2(2x + 1)\) has a solution at \(x = \frac{10}{9}\), validating the statement that it has a solution. If you need further explanations or more examples, please let me know!